三维多尺度离散Radon与John变换
Three-dimensional multiscale discrete Radon and John transforms
AI总结：
本文提出三维离散Radon与John变换的多尺度算法，无需插值，仅用整数运算实现线性对数复杂度，并证明其快速精确逆变换，适用于实时三维体数据处理。
摘要：
本文提出了两种算法，用于在离散三维体数据中采用多尺度方法计算离散积分变换，并考虑了其实时实现。第一种算法称为三维离散Radon变换（DRT）的平面变换，用于计算立方体中离散平面上的数值之和，该变换在离散数据中模拟了连续Radon变换在三维体数据中二维平面上的积分。这些平面的法线在上升方向上等间距分布，覆盖一个四边形化的半球，并包含12个十二分角。第二种提出的算法称为三维离散John变换（DJT）的线变换，用于对离散三维线上的元素求和，模拟John或X射线连续变换在三维体数据上的行为。这些离散积分变换不对输入或中间数据进行插值，且仅使用整数运算即可计算，复杂度为线性对数级；因此，在实时应用中优于基于傅里叶切片投影定理的方法。我们简要证明了这些变换具有精确的快速逆变换算法，适用于离散输入。
英文摘要：
Two algorithms are introduced for the computation of discrete integral transforms with a multiscale approach operating in discrete three-dimensional (3D) volumes while considering its real-time implementation. The first algorithm, referred to as 3D discrete Radon transform (DRT) of planes, will compute the summation set of values lying in discrete planes in a cube that imitates, in discrete data, the integrals on two-dimensional planes in a 3D volume similar to the continuous Radon transform. The normals of these planes, equispaced in ascents, cover a quadrilateralized hemisphere and comprise 12 dodecants. The second proposed algorithm, referred to as the 3D discrete John transform (DJT) of lines, will sum elements lying on discrete 3D lines while imitating the behavior of the John or X-ray continuous transform on 3D volumes. These discrete integral transforms do not perform interpolation on input or intermediate data, and they can be computed using only integer arithmetics with linearithmic complexity; thus, outperforming the methods based on the Fourier slice-projection theorem for real-time applications. We briefly prove that these transforms have fast inversion algorithms that are exact for discrete inputs.
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